Abstract

Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems of 𝑛 ordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity: ̇𝑧(𝑡)=𝐴𝑧(𝑡−𝜏)+𝑔(𝑡)+𝜀𝑍(𝑧(ℎ𝑖(𝑡),𝑡,𝜀),𝑡∈[𝑎,𝑏], assuming that these solutions satisfy the initial and boundary conditions 𝑧(𝑠)∶=𝜓(𝑠)if𝑠∉[𝑎,𝑏],ℓ𝑧(⋅)=𝛼∈ℝ𝑚. The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functional ℓ) does not coincide with the number of unknowns in the differential system with a single delay.

1. Introduction

First, we derive some auxiliary results concerning the theory of differential equations with delay. Consider a system of linear differential equations with concentrated delaẏ𝑧(𝑡)−𝐴(𝑡)𝑧ℎ0(𝑡)=𝑔(𝑡)if𝑡∈[𝑎,𝑏],(1.1)
assuming that𝑧(𝑠)∶=𝜓(𝑠)if𝑠∉[𝑎,𝑏],(1.2)
where 𝐴 is an 𝑛×𝑛 real matrix and 𝑔 is an 𝑛-dimensional real column-vector with components in the space 𝐿𝑝[𝑎,𝑏] (where 𝑝∈[1,∞)) of functions summable on [𝑎,𝑏]; the delay ℎ0(𝑡)≤𝑡 is a function ℎ0∶[𝑎,𝑏]→ℝ measurable on [𝑎,𝑏]; 𝜓∶ℝ⧵[𝑎,𝑏]→ℝ𝑛 is a given function. Using the denotations𝑆ℎ0𝑧(𝑡)∶=⎧⎪⎨⎪⎩𝑧ℎ0(𝑡)ifℎ0(𝑡)∈[𝑎,𝑏],𝜃ifℎ0(𝑡)∉[𝑎,𝑏],(1.3)𝜓ℎ0(𝑡)∶=⎧⎪⎨⎪⎩𝜃ifℎ0(𝑡)∈[𝑎,𝑏],𝜓ℎ0(𝑡)ifℎ0(𝑡)∉[𝑎,𝑏],(1.4)
where 𝜃 is an 𝑛-dimensional zero column-vector and assuming 𝑡∈[𝑎,𝑏], it is possible to rewrite (1.1), (1.2) as(𝐿𝑧)(𝑡)∶=̇𝑧(𝑡)−𝐴(𝑡)𝑆ℎ0𝑧(𝑡)=𝜑(𝑡),𝑡∈[𝑎,𝑏],(1.5)
where 𝜑 is an 𝑛-dimensional column-vector defined by the formula𝜑(𝑡)∶=𝑔(𝑡)+𝐴(𝑡)𝜓ℎ0(𝑡)∈𝐿𝑝[𝑎,𝑏].(1.6)
We will investigate (1.5) assuming that the operator 𝐿 maps a Banach space 𝐷𝑝[𝑎,𝑏] of absolutely continuous functions 𝑧∶[𝑎,𝑏]→ℝ𝑛 into a Banach space 𝐿𝑝[𝑎,𝑏](1≤𝑝<∞) of functions 𝜑∶[𝑎,𝑏]→ℝ𝑛 summable on [𝑎,𝑏]; the operator 𝑆ℎ0 maps the space 𝐷𝑝[𝑎,𝑏] into the space 𝐿𝑝[𝑎,𝑏]. Transformations (1.3), (1.4) make it possible to add the initial function 𝜓(𝑠), 𝑠<𝑎 to nonhomogeneity generating an additive and homogeneous operation not depending on 𝜓 and without the classical assumption regarding the continuous connection of solution 𝑧(𝑡) with the initial function 𝜓(𝑡) at the point 𝑡=𝑎.

A solution of differential system (1.5) is defined as an 𝑛-dimensional column vector-function 𝑧∈𝐷𝑝[𝑎,𝑏], absolutely continuous on [𝑎,𝑏], with a derivative ̇𝑧∈𝐿𝑝[𝑎,𝑏] satisfying (1.5) almost everywhere on [𝑎,𝑏].

Such approach makes it possible to apply well-developed methods of linear functional analysis to (1.5) with a linear and bounded operator 𝐿. It is well-known (see: [1, 2]) that a nonhomogeneous operator equation (1.5) with delayed argument is solvable in the space 𝐷𝑝[𝑎,𝑏] for an arbitrary right-hand side 𝜑∈𝐿𝑝[𝑎,𝑏] and has an 𝑛-dimensional family of solutions (dimker𝐿=𝑛) in the form𝑧(𝑡)=𝑋(𝑡)𝑐+𝑏𝑎𝐾(𝑡,𝑠)𝜑(𝑠)𝑑𝑠∀𝑐∈ℝ𝑛,(1.7)
where the kernel 𝐾(𝑡,𝑠) is an 𝑛×𝑛 Cauchy matrix defined in the square [𝑎,𝑏]×[𝑎,𝑏] being, for every fixed 𝑠≤𝑡, a solution of the matrix Cauchy problem (𝐿𝐾(⋅,𝑠))(𝑡)∶=𝜕𝐾(𝑡,𝑠)𝜕𝑡−𝐴(𝑡)𝑆ℎ0𝐾(⋅,𝑠)(𝑡)=Θ,𝐾(𝑠,𝑠)=𝐼,(1.8)
where 𝐾(𝑡,𝑠)≡Θ if 𝑎≤𝑡<𝑠≤𝑏, Θ is 𝑛×𝑛 null matrix and 𝐼 is 𝑛×𝑛 identity matrix. A fundamental 𝑛×𝑛 matrix 𝑋(𝑡) for the homogeneous (𝜑≡𝜃) equation (1.5) has the form 𝑋(𝑡)=𝐾(𝑡,𝑎), 𝑋(𝑎)=𝐼 [2]. Throughout the paper, we denote by Θ𝑠 an 𝑠×𝑠 null matrix if 𝑠≠𝑛, by Θ𝑠,𝑝 an 𝑠×𝑝 null matrix, by 𝐼𝑠 an 𝑠×𝑠 identity matrix if 𝑠≠𝑛, and by 𝜃𝑠 an 𝑠-dimensional zero column-vector if 𝑠≠𝑛.

A serious disadvantage of this approach, when investigating the above-formulated problem, is the necessity to find the Cauchy matrix 𝐾(𝑡,𝑠) [3, 4]. It exists but, as a rule, can only be found numerically. Therefore, it is important to find systems of differential equations with delay such that this problem can be solved directly. Below we consider the case of a system with so-called single delay [5]. In this case, the problem of how to construct the Cauchy matrix is successfully solved analytically due to a delayed matrix exponential defined below.

1.1. A Delayed Matrix Exponential

Consider a Cauchy problem for a linear nonhomogeneous differential system with constant coefficients and with a single delay 𝜏̇𝑧(𝑡)=𝐴𝑧(𝑡−𝜏)+𝑔(𝑡),(1.9)𝑧(𝑠)=𝜓(𝑠),if𝑠∈[−𝜏,0],(1.10)
with an 𝑛×𝑛 constant matrix 𝐴, 𝑔∶[0,∞)→ℝ𝑛, 𝜓∶[−𝜏,0]→ℝ𝑛, 𝜏>0 and an unknown vector-solution 𝑧∶[−𝜏,∞)→ℝ𝑛. Together with a nonhomogeneous problem (1.9), (1.10), we consider a related homogeneous problem ̇𝑧(𝑡)=𝐴𝑧(𝑡−𝜏),𝑧(𝑠)=𝜓(𝑠),if𝑠∈[−𝜏,0].(1.11)

Denote by 𝑒𝐴𝑡𝜏 a matrix function called a delayed matrix exponential (see [5]) and defined as𝑒𝐴𝑡𝜏∶=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩Θif−∞<𝑡<−𝜏,𝐼if−𝜏≤𝑡<0,𝐼+𝐴𝑡1!if0≤𝑡<𝜏,𝐼+𝐴𝑡1!+𝐴2(𝑡−𝜏)22!if𝜏≤𝑡<2𝜏,⋯𝐼+𝐴𝑡1!+⋯+𝐴𝑘(𝑡−(𝑘−1)𝜏)𝑘𝑘!if(𝑘−1)𝜏≤𝑡<𝑘𝜏,⋯.(1.12)
This definition can be reduced to the following expression: 𝑒𝐴𝑡𝜏=[𝑡/𝜏]+1𝑛=0𝐴𝑛(𝑡−(𝑛−1)𝜏)𝑛𝑛!,(1.13)
where [𝑡/𝜏] is the greatest integer function. The delayed matrix exponential equals the unit matrix 𝐼 on [−𝜏,0] and represents a fundamental matrix of a homogeneous system with single delay. Thus, the delayed matrix exponential solves the Cauchy problem for a homogeneous system (1.11), satisfying the unit initial conditions 𝑧(𝑠)=𝜓(𝑠)≡𝑒𝐴𝑠𝜏=𝐼if−𝜏≤𝑠≤0,(1.14)
and the following statement holds (see, e.g., [5], [6, Remark 1], [7, Theorem 2.1]).

Lemma 1.1. A solution of a Cauchy problem for a nonhomogeneous system with single delay (1.9), satisfying a constant initial condition
𝑧(𝑠)=𝜓(𝑠)=𝑐∈ℝ𝑛if𝑠∈[−𝜏,0](1.15)
has the form
𝑧(𝑡)=𝑒𝐴(𝑡−𝜏)𝜏𝑐+𝑡0𝑒𝐴(𝑡−𝜏−𝑠)𝜏𝑔(𝑠)𝑑𝑠.(1.16)

The delayed matrix exponential was applied, for example, in [6, 7] to investigation of boundary value problems of diffferential systems with a single delay and in [8] to investigation of the stability of linear perturbed systems with a single delay.

1.2. Fredholm Boundary-Value Problem

Without loss of generality, let 𝑎=0 and, with a view of the above, the problem (1.9), (1.10) can be transformed (ℎ0(𝑡)∶=𝑡−𝜏) to an equation of the type (1.1) (see (1.5))̇𝑧(𝑡)−𝐴𝑆ℎ0𝑧(𝑡)=𝜑(𝑡),𝑡∈[0,𝑏],(1.17)
where, in accordance with (1.3),(1.4)𝑆ℎ0𝑧(𝑡)=⎧⎪⎨⎪⎩𝑧(𝑡−𝜏)if𝑡−𝜏∈[0,𝑏],𝜃if𝑡−𝜏∉[0,𝑏],𝜑(𝑡)=𝑔(𝑡)+𝐴𝜓ℎ0(𝑡)∈𝐿𝑝[0,𝑏],𝜓ℎ0(𝑡)=⎧⎪⎨⎪⎩𝜃if𝑡−𝜏∈[0,𝑏],𝜓(𝑡−𝜏)if𝑡−𝜏∉[0,𝑏].(1.18)

A general solution of problem (1.17) for a nonhomogeneous system with single delay and zero initial data has the form (1.7) 𝑧(𝑡)=𝑋(𝑡)𝑐+𝑏0𝐾(𝑡,𝑠)𝜑(𝑠)𝑑𝑠∀𝑐∈ℝ𝑛,(1.19)
where, as can easily be verified (in view of the above-defined delayed matrix exponential) by substituting into (1.17), 𝑋(𝑡)=𝑒𝐴(𝑡−𝜏)𝜏,𝑋(0)=𝑒−𝐴𝜏𝜏=𝐼(1.20)
is a normal fundamental matrix of the homogeneous system related to (1.17) (or (1.9)) with initial data 𝑋(0)=𝐼, and the Cauchy matrix 𝐾(𝑡,𝑠) has the form 𝐾(𝑡,𝑠)=𝑒𝐴(𝑡−𝜏−𝑠)𝜏if0≤𝑠<𝑡≤𝑏,𝐾(𝑡,𝑠)≡Θif0≤𝑡<𝑠≤𝑏.(1.21)
Obviously 𝐾(𝑡,0)=𝑒𝐴(𝑡−𝜏)𝜏=𝑋(𝑡),𝐾(0,0)=𝑒𝐴(−𝜏)𝜏=𝑋(0)=𝐼,(1.22)
and, therefore, the initial problem (1.17) for systems of ordinary differential equations with constant coefficients and single delay has an 𝑛-parametric family of linearly independent solutions (1.16).

Now, we will deal with a general boundary-value problem for system (1.17). Using the results [2, 9], it is easy to derive statements for a general boundary-value problem if the number 𝑚 of boundary conditions does not coincide with the number 𝑛 of unknowns in a differential system with single delay.

We consider a boundary-value problem ̇𝑧(𝑡)−𝐴𝑧(𝑡−𝜏)=𝑔(𝑡),𝑡∈[0,𝑏],𝑧(𝑠)∶=𝜓(𝑠),𝑠∉[0,𝑏],(1.23)
assuming that ℓ𝑧(⋅)=𝛼∈ℝ𝑚,(1.24)
or, using (1.18), its equivalent forṁ𝑧(𝑡)−𝐴𝑆ℎ0𝑧(𝑡)=𝜑(𝑡),𝑡∈[0,𝑏],ℓ𝑧(⋅)=𝛼∈ℝ𝑚,(1.25)
where 𝛼 is an 𝑚-dimensional constant vector-column ℓ is an 𝑚-dimensional linear vector-functional defined on the space 𝐷𝑝[0,𝑏] of an 𝑛-dimensional vector-functions ℓ=colℓ1,…,ℓ𝑚∶𝐷𝑝[0,𝑏]⟶ℝ𝑚,ℓ𝑖∶𝐷𝑝[0,𝑏]⟶ℝ,𝑖=1,…,𝑚,(1.26)
absolutely continuous on [0,𝑏]. Such problems for functional-differential equations are of Fredholm's type (see, e.g., [1, 2]). In order to formulate the following result, we need several auxiliary abbreviations. We set 𝑄∶=ℓ𝑋(⋅)=ℓ𝑒𝐴(⋅−𝜏)𝜏.(1.27)
We define an 𝑛×𝑛-dimensional matrix (orthogonal projection) 𝑃𝑄∶=𝐼−𝑄+𝑄,(1.28)
projecting space ℝ𝑛 to ker𝑄 of the matrix 𝑄.

Moreover, we define an 𝑚×𝑚-dimensional matrix (orthogonal projection) 𝑃𝑄∗∶=𝐼𝑚−𝑄𝑄+,(1.29)
projecting space ℝ𝑚 to ker𝑄∗ of the transposed matrix 𝑄∗=𝑄𝑇, where 𝐼𝑚 is an 𝑚×𝑚 identity matrix and 𝑄+ is an 𝑛×𝑚-dimensional matrix pseudoinverse to the 𝑚×𝑛-dimensional matrix 𝑄. Denote 𝑑∶=rank𝑃𝑄∗ and 𝑛1∶=rank𝑄=rank𝑄∗. Since rank𝑃𝑄∗=𝑚−rank𝑄∗,(1.30)
we have 𝑑=𝑚−𝑛1.

We will denote by 𝑃𝑄∗𝑑 an 𝑑×𝑚-dimensional matrix constructed from 𝑑 linearly independent rows of the matrix 𝑃𝑄∗. Denote 𝑟∶=rank𝑃𝑄. Since rank𝑃𝑄=𝑛−rank𝑄,(1.31)
we have 𝑟=𝑛−𝑛1. By 𝑃𝑄𝑟 we will denote an 𝑛×𝑟-dimensional matrix constructed from 𝑟 linearly independent columns of the matrix 𝑃𝑄. Finally, we define 𝑋𝑟(𝑡)∶=𝑋(𝑡)𝑃𝑄𝑟,(1.32)
and a generalized Green operator (𝐺𝜑)(𝑡)∶=𝑏0𝐺(𝑡,𝑠)𝜑(𝑠)𝑑𝑠,(1.33)
where 𝐺(𝑡,𝑠)∶=𝐾(𝑡,𝑠)−𝑒𝐴(𝑡−𝜏)𝜏𝑄+ℓ𝐾(⋅,𝑠)(1.34)
is a generalized Green matrix corresponding to the boundary-value problem (1.25) (the Cauchy matrix 𝐾(𝑡,𝑠) has the form (1.21)).

In [6, Theorem 4], the following result (formulating the necessary and sufficient conditions of solvability and giving representations of the solutions 𝑧∈𝐷𝑝[0,𝑏], ̇𝑧∈𝐿𝑝[0,𝑏] of the boundary-value problem (1.25) in an explicit analytical form) is proved.

Theorem 1.2. If 𝑛1≤min(𝑚,𝑛), then: (i)the homogeneous problem
̇𝑧(𝑡)−𝐴𝑆ℎ0𝑧(𝑡)=𝜃,𝑡∈[0,𝑏],ℓ𝑧(⋅)=𝜃𝑚∈ℝ𝑚(1.35)
corresponding to problem (1.25) has exactly 𝑟 linearly independent solutions
𝑧𝑡,𝑐𝑟=𝑋𝑟(𝑡)𝑐𝑟=𝑒𝐴(𝑡−𝜏)𝜏𝑃𝑄𝑟𝑐𝑟∈𝐷𝑝[0,𝑏],(1.36)(ii)nonhomogeneous problem (1.25) is solvable in the space 𝐷𝑝[0,𝑏] if and only if 𝜑∈𝐿𝑝[0,𝑏] and 𝛼∈ℝ𝑚 satisfy 𝑑 linearly independent conditions
𝑃𝑄∗𝑑⋅𝛼−ℓ𝑏0𝐾(⋅,𝑠)𝜑(𝑠)𝑑𝑠=𝜃𝑑,(1.37)(iii)in that case the nonhomogeneous problem (1.25) has an 𝑟-dimensional family of linearly independent solutions represented in an analytical form
𝑧(𝑡)=𝑧0𝑡,𝑐𝑟∶=𝑋𝑟(𝑡)𝑐𝑟+(𝐺𝜑)(𝑡)+𝑋(𝑡)𝑄+𝛼∀𝑐𝑟∈ℝ𝑟.(1.38)

2. Perturbed Weakly Nonlinear Boundary Value Problems

As an example of applying Theorem 1.2, we consider a problem of the branching of solutions 𝑧∶[0,𝑏]→ℝ𝑛, 𝑏>0 of systems of nonlinear ordinary differential equations with a small parameter 𝜀 and with a finite number of measurable delays ℎ𝑖(𝑡), 𝑖=1,2,…,𝑘 of argument of the forṁ𝑧(𝑡)=𝐴𝑧(𝑡−𝜏)+𝑔(𝑡)+𝜀𝑍𝑧ℎ𝑖(𝑡),𝑡,𝜀,𝑡∈[0,𝑏],ℎ𝑖(𝑡)≤𝑡,(2.1)
satisfying the initial and boundary conditions𝑧(𝑠)=𝜓(𝑠),if𝑠<0,ℓ𝑧(⋅)=𝛼,𝛼∈ℝ𝑚,(2.2)
and such that its solution 𝑧=𝑧(𝑡,𝜀), satisfying 𝑧(⋅,𝜀)∈𝐷𝑝[0,𝑏],̇𝑧(⋅,𝜀)∈𝐿𝑝[0,𝑏],𝑧(𝑡,⋅)∈𝐶0,𝜀0,(2.3)
for a sufficiently small 𝜀0>0, for 𝜀=0, turns into one of the generating solutions (1.38); that is, 𝑧(𝑡,0)=𝑧0(𝑡,𝑐𝑟) for a 𝑐𝑟∈ℝ𝑟. We assume that the 𝑛×1 vector-operator 𝑍 satisfies 𝑍(⋅,𝑡,𝜀)∈𝐶1‖‖𝑧−𝑧0‖‖≤𝑞,𝑍𝑧ℎ𝑖(𝑡),⋅,𝜀∈𝐿𝑝[0,𝑏],𝑍𝑧ℎ𝑖(𝑡),𝑡,⋅∈𝐶0,𝜀0,(2.4)
where 𝑞>0 is sufficiently small. Using denotations (1.3), (1.4), and (1.6), it is easy to show that the perturbed nonlinear boundary value problem (2.1), (2.2) can be rewritten in the forṁ𝑧(𝑡)=𝐴𝑆ℎ0𝑧(𝑡)+𝜀𝑍𝑆ℎ𝑧(𝑡),𝑡,𝜀+𝜑(𝑡),ℓ𝑧(⋅)=𝛼,𝑡∈[0,𝑏].(2.5)
In (2.5), 𝐴 is an 𝑛×𝑛 constant matrix, ℎ0∶[0,𝑏]→ℝ is a single delay defined by ℎ0(𝑡)∶=𝑡−𝜏, 𝜏>0, 𝑆ℎ𝑧(𝑡)=col𝑆ℎ1𝑧(𝑡),…,𝑆ℎ𝑘𝑧(𝑡)(2.6)
is an 𝑁-dimensional column vector, where 𝑁=𝑛𝑘, and 𝜑 is an 𝑛-dimensional column vector given by 𝜑(𝑡)=𝑔(𝑡)+𝐴𝜓ℎ0(𝑡).(2.7)
The operator 𝑆ℎ maps the space 𝐷𝑝 into the space 𝐿𝑁𝑝=𝐿𝑝×⋯×𝐿𝑝𝑘-times,(2.8)
that is, 𝑆ℎ∶𝐷𝑝→𝐿𝑁𝑝. Using denotation (1.3) for the operator 𝑆ℎ𝑖∶𝐷𝑝→𝐿𝑝, 𝑖=1,…,𝑘, we have the following representation:𝑆ℎ𝑖𝑧(𝑡)=𝑏0𝜒ℎ𝑖(𝑡,𝑠)̇𝑧(𝑠)𝑑𝑠+𝜒ℎ𝑖(𝑡,0)𝑧(0),(2.9)
where 𝜒ℎ𝑖(𝑡,𝑠)=⎧⎪⎨⎪⎩1,if(𝑡,𝑠)∈Ω𝑖,0,if(𝑡,𝑠)∉Ω𝑖(2.10)
is the characteristic function of the set Ω𝑖∶=(𝑡,𝑠)∈[0,𝑏]×[0,𝑏]∶0≤𝑠≤ℎ𝑖(𝑡)≤𝑏.(2.11)
Assume that the generating boundary value probleṁ𝑧(𝑡)=𝐴𝑆ℎ0𝑧(𝑡)+𝜑(𝑡),𝑙𝑧=𝛼,(2.12)
being a particular case of (2.5) for 𝜀=0, has solutions for nonhomogeneities 𝜑∈𝐿𝑝[0,𝑏] and 𝛼∈ℝ𝑚 that satisfy conditions (1.37). In such a case, by Theorem 1.2, the problem (2.12) possesses an 𝑟-dimensional family of solutions of the form (1.38).

Problem 1. Below, we consider the following problem: derive the necessary and sufficient conditions indicating when solutions of (2.5) turn into solutions (1.38) of the boundary value problem (2.12) for 𝜀=0.

Using the theory of generalized inverse operators [2], it is possible to find conditions for the solutions of the boundary value problem (2.5) to be branching from the solutions of (2.5) with 𝜀=0. Below, we formulate statements, solving the above problem. As compared with an earlier result [10, page 150], the present result is derived in an explicit analytical form. The progress was possible by using the delayed matrix exponential since, in such a case, all the necessary calculations can be performed to the full.

Theorem 2.1 (necessary condition). Consider the system (2.1); that is,
̇𝑧(𝑡)=𝐴𝑧(𝑡−𝜏)+𝑔(𝑡)+𝜀𝑍𝑧ℎ𝑖(𝑡),𝑡,𝜀,𝑡∈[0,𝑏],(2.13)
where ℎ𝑖(𝑡)≤𝑡, 𝑖=1,…,𝑘, with the initial and boundary conditions (2.2); that is,
𝑧(𝑠)=𝜓(𝑠),if𝑠<0<𝑏,ℓ𝑧(⋅)=𝛼∈ℝ𝑚,(2.14)
and assume that, for nonhomogeneities
𝜑(𝑡)=𝑔(𝑡)+𝐴𝜓ℎ0(𝑡)∈𝐿𝑝[0,𝑏],(2.15)
and for 𝛼∈ℝ𝑚, the generating boundary value problem
̇𝑧(𝑡)=𝐴𝑆ℎ0𝑧(𝑡)+𝜑(𝑡),ℓ𝑧(⋅)=𝛼,(2.16)
corresponding to the problem (1.25), has exactly an 𝑟-dimensional family of linearly independent solutions of the form (1.38). Moreover, assume that the boundary value problem (2.13), (2.14) has a solution 𝑧(𝑡,𝜀) which, for 𝜀=0, turns into one of solutions 𝑧0(𝑡,𝑐𝑟) in (1.38) with a vector-constant 𝑐𝑟∶=𝑐0𝑟∈ℝ𝑟. Then, the vector 𝑐0𝑟 satisfies the equation 𝐹𝑐0𝑟∶=𝑏0𝐻(𝑠)𝑍𝑆ℎ𝑧0𝑠,𝑐0𝑟,𝑠,0𝑑𝑠=𝜃𝑑,(2.17)
where
𝐻(𝑠)∶=𝑃𝑄∗𝑑ℓ𝐾(⋅,𝑠)=𝑃𝑄∗𝑑ℓ𝑒𝐴(⋅−𝜏−𝑠)𝜏.(2.18)

Proof. We consider the nonlinearity in system (2.13), that is, the term 𝜀𝑍(𝑧(ℎ𝑖(𝑡)),𝑡,𝜀) as an inhomogeneity, and use Theorem 1.2 assuming that condition (1.37) is satisfied. This gives
𝑏0𝐻(𝑠)𝑍𝑆ℎ𝑧(𝑠,𝜀),𝑠,𝜀𝑑𝑠=𝜃𝑑.(2.19)
In this integral, letting 𝜀→0, we arrive at the required condition (2.17).

Corollary 2.2. For periodic boundary-value problems, the vector-constant 𝑐𝑟∈ℝ𝑟 has a physical meaning-it is the amplitude of the oscillations generated. For this reason, (2.17) is called an equation generating the amplitude [11]. By analogy with the investigation of periodic problems, it is natural to say (2.17) is an equation for generating the constants of the boundary value problem (2.13), (2.14). If (2.17) is solvable, then the vector constant 𝑐0𝑟∈ℝ𝑟 specifies the generating solution 𝑧0(𝑡,𝑐0𝑟) corresponding to the solution 𝑧=𝑧(𝑡,𝜀) of the original problem such that
𝑧(⋅,𝜀)∶[0,𝑏]⟶ℝ𝑛,𝑧(⋅,𝜀)∈𝐷𝑝[0,𝑏],̇𝑧(⋅,𝜀)∈𝐿𝑝[0,𝑏],𝑧(𝑡,⋅)∈𝐶0,𝜀0,𝑧(𝑡,0)=𝑧0𝑡,𝑐0𝑟.(2.20)
Also, if (2.17) is unsolvable, the problem (2.13), (2.14) has no solution in the analyzed space. Note that, here and in what follows, all expressions are obtained in the real form and hence, we are interested in real solutions of (2.17), which can be algebraic or transcendental.

Sufficient conditions for the existence of solutions of the boundary-value problem (2.13), (2.14) can be derived using results in [10, page 155] and [2]. By changing the variables in system (2.13), (2.14) 𝑧(𝑡,𝜀)=𝑧0𝑡,𝑐0𝑟+𝑦(𝑡,𝜀),(2.21)
we arrive at a problem of finding sufficient conditions for the existence of solutions of the probleṁ𝑦(𝑡)=𝐴𝑆ℎ0𝑦(𝑡)+𝜀𝑍𝑆ℎ𝑧0+𝑦(𝑡),𝑡,𝜀,ℓ𝑦=𝜃𝑚,𝑡∈[0,𝑏],(2.22)
and such that 𝑦(⋅,𝜀)∶[0,𝑏]⟶ℝ𝑛,𝑦(⋅,𝜀)∈𝐷𝑝[0,𝑏],̇𝑦(⋅,𝜀)∈𝐿𝑝[0,𝑏],𝑦(𝑡,⋅)∈𝐶0,𝜀0,𝑦(𝑡,0)=𝜃.(2.23)
Since the vector function 𝑍((𝑆ℎ𝑧)(𝑡),𝑡,𝜀) is continuously differentiable with respect to 𝑧 and continuous in 𝜀 in the neighborhood of the point (𝑧,𝜀)=𝑧0𝑡,𝑐0𝑟,0,(2.24)
we can separate its linear term as a function depending on 𝑦 and terms of order zero with respect to 𝜀𝑍𝑆ℎ𝑧0𝑡,𝑐0𝑟+𝑦,𝑡,𝜀=𝑓0𝑡,𝑐0𝑟+𝐴1(𝑡)𝑆ℎ𝑦(𝑡)+𝑅𝑆ℎ𝑦(𝑡),𝑡,𝜀,(2.25)
where 𝑓0𝑡,𝑐0𝑟∶=𝑍𝑆ℎ𝑧0𝑡,𝑐0𝑟,𝑡,0,𝑓0⋅,𝑐0𝑟∈𝐿𝑝[0,𝑏],𝐴1(𝑡)=𝐴1𝑡,𝑐0𝑟=𝜕𝑍(𝑆ℎ𝑥,𝑡,0)𝜕𝑆ℎ𝑥||||𝑥=𝑧0(𝑡,𝑐0𝑟),𝐴1(⋅)∈𝐿𝑝[0,𝑏],𝑅(𝜃,𝑡,0)=𝜃,𝜕𝑅(𝜃,𝑡,0)𝜕𝑦=Θ,𝑅(𝑦,⋅,𝜀)∈𝐿𝑝[0,𝑏].(2.26)
We now consider the vector function 𝑍((𝑆ℎ(𝑧0+𝑦))(𝑡),𝑡,𝜀) in (2.22) as an inhomogeneity and we apply Theorem 1.2 to this system. As the result, we obtain the following representation for the solution of (2.22): 𝑦(𝑡,𝜀)=𝑋𝑟(𝑡)𝑐+𝑦(1)(𝑡,𝜀).(2.27)
In this expression, the unknown vector of constants 𝑐=𝑐(𝜀)∈𝐶[0,𝜀0] is determined from a condition similar to condition (1.37) for the existence of solution of problem (2.22):𝐵0𝑐=𝑏0𝐻(𝑠)𝐴1(𝑠)𝑆ℎ𝑦(1)(𝑠,𝜀)+𝑅𝑆ℎ𝑦(𝑠,𝜀),𝑠,𝜀𝑑𝑠,(2.28)
where 𝐵0=𝑏0𝐻(𝑠)𝐴1(𝑠)𝑆ℎ𝑋𝑟(𝑠)𝑑𝑠(2.29)
is a 𝑑×𝑟 matrix, and 𝐻(𝑠)∶=𝑃𝑄∗𝑑ℓ𝐾(⋅,𝑠)=𝑃𝑄∗𝑑ℓ𝑒𝐴(⋅−𝜏−𝑠)𝜏.(2.30)
The unknown vector function 𝑦(1)(𝑡,𝜀) is determined by using the generalized Green operator as follows: 𝑦(1)(𝑡,𝜀)=𝜀𝐺𝑍𝑆ℎ𝑧0𝑠,𝑐0𝑟+𝑦,𝑠,𝜀(𝑡).(2.31)
Let 𝑃𝑁(𝐵0) be an 𝑟×𝑟 matrix orthoprojector ℝ𝑟→𝑁(𝐵0), and let 𝑃𝑁(𝐵∗0) be a 𝑑×𝑑 matrix-orthoprojector ℝ𝑑→𝑁(𝐵∗0). Equation (2.28) is solvable with respect to 𝑐∈ℝ𝑟 if and only if 𝑃𝑁(𝐵∗0)𝑏0𝐻(𝑠)𝐴1(𝑠)𝑆ℎ𝑦(1)(𝑠,𝜀)+𝑅𝑆ℎ𝑦(𝑠,𝜀),𝑠,𝜀𝑑𝑠=𝜃𝑑.(2.32)
For 𝑃𝑁(𝐵∗0)=Θ𝑑,(2.33)
the last condition is always satisfied and (2.28) is solvable with respect to 𝑐∈ℝ𝑟 up to an arbitrary vector constant 𝑃𝑁(𝐵0)𝑐∈ℝ𝑟 from the null space of the matrix 𝐵0𝑐=𝐵+0𝑏0𝐻(𝑠)𝐴1(𝑠)𝑆ℎ𝑦(1)(𝑠,𝜀)+𝑅𝑆ℎ𝑦(𝑠,𝜀),𝑠,𝜀𝑑𝑠+𝑃𝑁(𝐵0)𝑐.(2.34)
To find a solution 𝑦=𝑦(𝑡,𝜀) of (2.28) such that 𝑦(⋅,𝜀)∶[0,𝑏]⟶𝑅𝑛,𝑦(⋅,𝜀)∈𝐷𝑝[0,𝑏],̇𝑦(⋅,𝜀)∈𝐿𝑝[0,𝑏],𝑦(𝑡,⋅)∈𝐶0,𝜀0,𝑦(𝑡,0)=𝜃,(2.35)
it is necessary to solve the following operator system:𝑦(𝑡,𝜀)=𝑋𝑟(𝑡)𝑐+𝑦(1)(𝑡,𝜀),𝑐=𝐵+0𝑏0𝐻(𝑠)𝐴1(𝑠)𝑆ℎ𝑦(1)(𝑠,𝜀)+𝑅𝑆ℎ𝑦(𝑠,𝜀),𝑠,𝜀𝑑𝑠,𝑦(1)(𝑡,𝜀)=𝜀𝐺𝑍𝑆ℎ𝑧0𝑠,𝑐0𝑟+𝑦,𝑠,𝜀(𝑡).(2.36)
The operator system (2.36) belongs to the class of systems solvable by the method of simple iterations, convergent for sufficiently small 𝜀∈[0,𝜀0] (see [10, page 188]). Indeed, system (2.36) can be rewritten in the form𝑢=𝐿(1)𝑢+𝐹𝑢,(2.37)
where 𝑢=col(𝑦(𝑡,𝜀),𝑐(𝜀),𝑦(1)(𝑡,𝜀)) is a (2𝑛+𝑟)-dimensional column vector, 𝐿(1) is a linear operator 𝐿(1)∶=⎛⎜⎜⎜⎝Θ𝑋𝑟(𝑡)𝐼Θ𝑟,𝑛Θ𝑟,𝑟𝐿1ΘΘ𝑛,𝑟Θ⎞⎟⎟⎟⎠,(2.38)
where 𝐿1(∗)=𝐵+0𝑏0𝐻(𝑠)𝐴1(𝑠)(∗)𝑑𝑠,(2.39)
and 𝐹 is a nonlinear operator 𝐹𝑢∶=⎛⎜⎜⎜⎜⎝𝜃𝐵+0𝑏0𝐻(𝑠)𝑅𝑆ℎ𝑦(𝑠,𝜀),𝑠,𝜀𝑑𝑠𝜀𝐺𝑍𝑆ℎ𝑧0𝑠,𝑐0𝑟,𝑠,0+𝐴1(𝑠)𝑆ℎ𝑦(𝑠,𝜀)+𝑅𝑆ℎ𝑦(𝑠,𝜀),𝑠,𝜀(𝑡)⎞⎟⎟⎟⎟⎠.(2.40)
In view of the structure of the operator 𝐿(1) containing zero blocks on and below the main diagonal, the inverse operator 𝐼2𝑛+𝑟−𝐿(1)−1(2.41)
exists. System (2.37) can be transformed into 𝑢=𝑆𝑢,(2.42)
where 𝑆∶=𝐼2𝑛+𝑟−𝐿(1)−1𝐹(2.43)
is a contraction operator in a sufficiently small neighborhood of the point (𝑧,𝜀)=𝑧0𝑡,𝑐0𝑟,0.(2.44)
Thus, the solvability of the last operator system can be established by using one of the existing versions of the fixed-point principles [12] applicable to the system for sufficiently small 𝜀∈[0,𝜀0]. It is easy to prove that the sufficient condition 𝑃𝑁(𝐵∗0)=Θ𝑑 for the existence of solutions of the boundary value problem (2.13), (2.14) means that the constant 𝑐0𝑟∈ℝ𝑟 of the equation for generating constant (2.17) is a simple root of equation (2.17) [2]. By using the method of simple iterations, we can find the solution of the operator system and hence the solution of the original boundary value problem (2.13), (2.14). Now, we arrive at the following theorem.

Theorem 2.3 (sufficient condition). Assume that the boundary value problem (2.13), (2.14) satisfies the conditions listed above and the corresponding linear boundary value problem (1.25) has an 𝑟-dimensional family of linearly independent solutions of the form (1.38). Then, for any simple root 𝑐𝑟=𝑐0𝑟∈ℝ𝑟 of the equation for generating the constants (2.17), there exist at least one solution of the boundary value problem (2.13), (2.14). The indicated solution 𝑧(𝑡,𝜀) is such that
𝑧(⋅,𝜀)∈𝐷𝑝[0,𝑏],̇𝑧(⋅,𝜀)∈𝐿𝑝[0,𝑏],𝑧(𝑡,⋅)∈𝐶0,𝜀0,(2.45)
and, for 𝜀=0, turns into one of the generating solutions (1.38) with a constant 𝑐0𝑟∈ℝ𝑟; that is, 𝑧(𝑡,0)=𝑧0(𝑡,𝑐0𝑟). This solution can be found by the method of simple iterations, which is convergent for a sufficiently small 𝜀∈[0,𝜀0].

Corollary 2.4. If the number 𝑛 of unknown variables is equal to the number 𝑚 of boundary conditions (and hence 𝑟=𝑑), the boundary value problem (2.13), (2.14) has a unique solution. In such a case, the problems considered for functional-differential equations are of Fredholm's type with a zero index. By using the procedure proposed in [2] with some simplifying assumptions, we can generalize the proposed method to the case of multiple roots of equation (2.17) to determine sufficient conditions for the existence of solutions of the boundary-value problem (2.13), (2.14).

3. Example

We will illustrate the above proved theorems on the example of a weakly perturbed linear boundary value problem. Consider the following simplest boundary value problem-a periodic problem for the delayed differential equation:̇𝑧(𝑡)=𝑧(𝑡−𝜏)+𝜀𝑘𝑖=1𝐵𝑖(𝑡)𝑧ℎ𝑖(𝑡)+𝑔(𝑡),𝑡∈(0,𝑇],𝑧(𝑠)=𝜓(𝑠),if𝑠<0,𝑧(0)=𝑧(𝑇),(3.1)
where 0<𝜏,𝑇=const, 𝐵𝑖 are 𝑛×𝑛 matrices, 𝐵𝑖,𝑔∈𝐿𝑝[0,𝑇], 𝜓∶ℝ1⧵(0,𝑇]→ℝ𝑛, ℎ𝑖(𝑡)≤𝑡 are measurable functions. Using the symbols 𝑆ℎ𝑖 and 𝜓ℎ𝑖 (see (1.3), (1.4), (2.9)), we arrive at the following operator system: ̇𝑧(𝑡)=𝑧(𝑡−𝜏)+𝜀𝐵(𝑡)𝑆ℎ𝑧(𝑡)+𝜑(𝑡,𝜀),ℓ𝑧∶=𝑧(0)−𝑧(𝑇)=𝜃𝑛,(3.2)
where 𝐵(𝑡)∶=(𝐵1(𝑡),…,𝐵𝑘(𝑡)) is an 𝑛×𝑁 matrix (𝑁=𝑛𝑘), and 𝜑(𝑡,𝜀)∶=𝑔(𝑡)+𝜓ℎ0(𝑡)+𝜀𝑘𝑖=1𝐵𝑖(𝑡)𝜓ℎ𝑖(𝑡)∈𝐿𝑝[0,𝑇].(3.3)
We will consider the simplest case with 𝑇≤𝜏. Utilizing the delayed matrix exponential, it can be easily verified that in this case, the matrix 𝑋(𝑡)=𝑒𝐼(𝑡−𝜏)𝜏=𝐼(3.4)
is a normal fundamental matrix for the homogeneous generating system ̇𝑧(𝑡)=𝑧(𝑡−𝜏).(3.5)
Then, 𝑄∶=ℓ𝑋(⋅)=𝑒−𝐼𝜏𝜏−𝑒𝐼(𝑇−𝜏)𝜏=𝜃𝑛,𝑃𝑄=𝑃𝑄∗=𝐼,(𝑟=𝑛,𝑑=𝑚=𝑛),𝐾(𝑡,𝑠)=⎧⎪⎨⎪⎩𝑒𝐼(𝑡−𝜏−𝑠)𝜏=𝐼,0≤𝑠≤𝑡≤𝑇,Θ,𝑠>𝑡,ℓ𝐾(⋅,𝑠)=𝐾(0,𝑠)−𝐾(𝑇,𝑠)=−𝐼,𝐻(𝜏)=𝑃𝑄∗ℓ𝐾(⋅,𝑠)=−𝐼,𝑆ℎ𝑖𝐼(𝑡)=𝜒ℎ𝑖(𝑡,0)⋅𝐼=𝐼⋅⎧⎪⎨⎪⎩1,if0≤ℎ𝑖(𝑡)≤𝑇,0,ifℎ𝑖(𝑡)<0.(3.6)
To illustrate the theorems proved above, we will find the conditions for which the boundary value problem (3.1) has a solution 𝑧(𝑡,𝜀) that, for 𝜀=0, turns into one of solutions (1.38) 𝑧0(𝑡,𝑐𝑟) of the generating problem. In contrast to the previous works [7, 9], we consider the case when the unperturbed boundary-value problem ̇𝑧(𝑡)=𝑧(𝑡−𝜏)+𝜑(𝑡,0),𝑧(0)=𝑧(𝑇)(3.7)
has an 𝑛-parametric family of linear-independent solutions of the form(1.38)𝑧∶=𝑧0𝑡,𝑐𝑛=𝑐𝑛+(𝐺𝜑)(𝑡),∀𝑐𝑛∈ℝ𝑛.(3.8)
For this purpose, it is necessary and sufficient for the vector function 𝜑(𝑡)=𝑔(𝑡)+𝜓ℎ0(𝑡)(3.9)
to satisfy the condition of type (1.37)𝑇0𝐻(𝑠)𝜑(𝑠)𝑑𝑠=−𝑇0𝜑(𝑠)𝑑𝑠=𝜃𝑛.(3.10)
Then, according to the Theorem 2.1, the constant 𝑐𝑛=𝑐0𝑛∈ℝ𝑛 must satisfy (2.17), that is, the equation𝐹𝑐0𝑛∶=𝑇0𝐻(𝑠)𝑍𝑆ℎ𝑧0𝑠,𝑐0𝑛,𝑠,0𝑑𝑠=𝜃𝑛,(3.11)
which in our case is a linear algebraic system 𝐵0𝑐0𝑛=−𝑇0𝐵(𝑠)𝑆ℎ(𝐺𝜑)(𝑠)𝑑𝑠,(3.12)
with the 𝑛×𝑛 matrix 𝐵0 in the form 𝐵0=𝑇0𝐻(𝑠)𝐵(𝑠)𝑆ℎ𝐼(𝑠)𝑑𝑠=−𝑇0𝑘𝑖=1𝐵𝑖(𝑠)𝑆ℎ𝑖𝐼(𝑠)𝑑𝑠=−𝑘𝑖=1𝑇0𝐵𝑖(𝑠)𝜒ℎ𝑖(𝑠,0)𝑑𝑠.(3.13)
According to Corollary 2.4, if det𝐵0≠0, the problem (3.1) for the case 𝑇≤𝜏 has a unique solution 𝑧(𝑡,𝜀) with the properties 𝑧(⋅,𝜀)∈𝐷𝑛𝑝[0,𝑇],̇𝑧(⋅,𝜀)∈𝐿𝑛𝑝[0,𝑇],𝑧(𝑡,⋅)∈𝐶0,𝜀0,𝑧(𝑡,0)=𝑧0𝑡,𝑐0𝑛,(3.14)
for 𝑔∈𝐿𝑝[0,𝑇], 𝜓(𝑡)∈𝐿𝑝[0,𝑇], and for measurable delays ℎ𝑖 that which satisfy the criterion (3.10) of the existence of a generating solution where 𝑐0𝑛=−𝐵+0𝑇0𝐵(𝑠)𝑆ℎ(𝐺𝜑)(𝑠)𝑑𝑠.(3.15)
A solution 𝑧(𝑡,𝜀) of the boundary value problem (3.1) can be found by the convergent method of simple iterations (see Theorem 2.3).

If, for example, ℎ𝑖(𝑡)=𝑡−Δ𝑖, where 0<Δ𝑖=const<𝑇, 𝑖=1,…,𝑘, then 𝜒ℎ𝑖(𝑡,0)=⎧⎪⎨⎪⎩1if0≤ℎ𝑖(𝑡)=𝑡−Δ𝑖≤𝑇,0ifℎ𝑖(𝑡)=𝑡−Δ𝑖<0,=⎧⎪⎨⎪⎩1ifΔ𝑖≤𝑡≤𝑇+Δ𝑖,0,if𝑡<Δ𝑖.(3.16)
The 𝑛×𝑛 matrix 𝐵0 can be rewritten in the form 𝐵0=𝑇0𝐻(𝑠)𝑘𝑖=1𝐵𝑖(𝑠)𝜒ℎ𝑖(𝑠,0)𝑑𝜏=−𝑘𝑖=1𝑇0𝐵𝑖(𝑠)𝜒ℎ𝑖(𝑠,0)𝑑𝑠=−𝑘𝑖=1𝑇Δ𝑖𝐵𝑖(𝑠)𝑑𝑠,(3.17)
and the unique solvability condition of the boundary value problem (3.1) takes the form det⎡⎢⎣𝑘𝑖=1𝑇Δ𝑖𝐵𝑖(𝑠)𝑑𝑠⎤⎥⎦≠0.(3.18)
It is easy to see that if the vector function 𝑍(𝑧(ℎ𝑖(𝑡)),𝑡,𝜀) is nonlinear in 𝑧, for example as a square, then (3.11) generating the constants will be a square-algebraic system and, in this case, the boundary value problem (3.1) can have two solutions branching from the point 𝜀=0.

Acknowledgments

The first and the fourth authors were supported by the Grant no. 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and by the project APVV-0700-07 of Slovak Research and Development Agency. The second author was supported by the Grant no. P201/11/0768 of Czech Grant Agency, by the Council of Czech Government MSM 0021630503 and by the Project FEKT/FSI-S-11-1-1159. The third author was supported by the Project no. M/34-2008 of Ukrainian Ministry of Education, Ukraine.